The Braid Index Of Reduced Alternating Links
Yuanan Diao, G\'abor Hetyei, Pengyu Liu

TL;DR
This paper characterizes when the braid index of an alternating link equals the number of Seifert circles in a reduced diagram, based on the properties of the Seifert graph, refining previous conjectures and results.
Contribution
It provides a precise criterion involving the Seifert graph for when the braid index matches the Seifert circles count in reduced diagrams.
Findings
Braid index equals Seifert circles if and only if the Seifert graph has no edges of weight one.
The result refines Murasugi's conjecture and connects to the MFW inequality and Yamada's theorem.
Characterization applies specifically to reduced alternating link diagrams.
Abstract
It is well known that the minimum crossing number of an alternating link equals the number of crossings in any reduced alternating link diagram of the link. This remarkable result is an application of the Jones polynomial. In the case of the braid index of an alternating link, Murasugi had conjectured that the number of Seifert circles in a reduced alternating diagram of the link equals the braid index of the link. This conjecture turned out to be false. In this paper we prove the next best thing that one could hope for: we characterize exactly those alternating links for which their braid indices equal to the numbers of Seifert circles in their corresponding reduced alternating link diagrams. More specifically, we prove that if is a reduced alternating link diagram of an alternating link , then , the braid index of , equals the number of Seifert circles in if and…
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Taxonomy
TopicsGeometric and Algebraic Topology · Advanced Combinatorial Mathematics · Homotopy and Cohomology in Algebraic Topology
